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\begin{document}

\preprint{ZAGREB-ZTF-02/01}

\title{On Selected Radiative Corrections to the Nondiagonal
Neutrino--Electron Interaction}% Force line breaks with \\

\author{Kre\v{s}imir Kumeri\v{c}ki}%
 \email{kkumer@phy.hr}
\author{Ivica Picek}%
 \email{picek@phy.hr}
\affiliation{%
Department of Physics, Faculty of Science, University of Zagreb,
 P.O.B. 331, HR-10002 Zagreb, Croatia
}%

\date{\today}%

\begin{abstract}
We present a contribution of higher order to neutrino--electron
scattering that is a charged-current counterpart of the anomalous
axial-vector triangle contribution. It arises in 
the standard model with massive neutrinos, and renormalizes
the axial-vector form-factor at low energies. We comment on some
conceivable implications for astrophysics and cosmology.
\end{abstract}

\pacs{11.15.-q, 13.10, 13.40.Ks, 14.60.Pq}% PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle

\section{Introduction}

 Ever since the earliest studies of the neutrino--electron scattering,
this particular process remained in a focus of interest. On the
experimental side it plays a major role in obtaining the recent results
at the SuperKamiokande detector \cite{SK98}.
The underlying \emph{diagonal} $\nu$--$e$ interaction
density is provided by the standard model (SM), which at energies much lower 
than $M_W$ gives
\begin{equation}
\mathcal{H}^{\text{diag}}(x)=\frac{G_F}{\sqrt{2}}\,\bar{e}\gamma^\mu 
 (g_V - g_A\gamma_5)
e\, \bar{\nu}\gamma_{\mu}(1-\gamma_5)\nu \;.
\label{ham}
\end{equation}
Here $g_V$ and $g_A$ are the SM couplings, where both
charged and neutral current contribute to
\begin{displaymath}
g_V  =  \frac{1}{2}+2\sin^{2}\theta_{W}\;,\;\; g_A=\frac{1}{2} 
\qquad\textrm{for}\qquad \nu=\nu_e \;,
\end{displaymath}
whereas there is only neutral current giving
\begin{displaymath}
g_V  =  -\frac{1}{2}+2\sin^{2}\theta_{W}\;,\;\; g_A=-\frac{1}{2}
 \qquad\textrm{for}\qquad\nu= \nu_\mu, \nu_\tau \;.
\end{displaymath}
The results for atmospheric neutrinos \cite{SK98}, in conjunction 
with the more recent results from the Sudbury
Neutrino Observatory (SNO) \cite{SNO01}, for the first time show
that the disappearance of one neutrino flavor (such as reported by
the SuperKamiokande)
is accompanied by the appearance of another.  Such conversion of the
neutrino species appears naturally if neutrinos have nonvanishing
masses.

The Hamiltonian density (\ref{ham}) represents a starting point in
numerous other considerations, such as a recent study of signals from the
cosmic neutrino background \cite{DuGN01} and the study of ionization of
atoms by keV neutrinos \cite{GuPP01}, to give some examples.
Let us stress here that at order $\alpha^2$ radiative corrections
from Fig.~\ref{triangle} generate an
infinite nonrenormalizable addition to the interaction (\ref{ham}).
Adler observed this infinite radiative correction to $\nu_{l}l$ scattering
by embedding the triangle diagram calculated by Rosenberg \cite{Ro63}
into the next loop. In present day terminology,
the axial triangle anomaly resides in the neutral current piece of the
$\nu$--$e$ interaction displayed on Fig.~\ref{triangle}(b).

\begin{figure}
\centerline{\includegraphics[scale=0.74]{triangle.eps}}
\vspace*{3ex}
\caption{Radiative corrections of order $\alpha^2$ displayed in
the 1PI blob (a), contain a nonrenormalizable triangle diagram
in the axial-vector part of the neutral current amplitude (b).
\label{triangle}}
\end{figure}

Among possible remedies for infinities proposed by Adler, the idea
to add
to the electron triangle on Fig.~\ref{triangle}(b) the muon triangle
with opposite sign, survived in its essence to the present day.
Miraculously, the quantum numbers assigned to the SM fermion
representations lead to such cancellation for each generation of fundamental
fermions.

In addition to this piece of the neutrino axial current for which nature
took care by itself, we can find another contribution at the
same order of $\alpha^2$, that is finite.
Up to our knowledge this extra part, to which we turn in this paper,
has not been presented in the literature.

In considering such counterpart to original Adler's contribution,
we allow for massive neutrinos. This will generalize the interaction (\ref{ham})
to the \emph{nondiagonal} heavy($\nu_H$)$\leftrightarrow$light($\nu_L$) transition
\begin{equation}
\mathcal{H}^{\text{non-diag}}_{\text{rad}}(x)=
\frac{G_F}{\sqrt{2}}\,\bar{e}\gamma^{\mu} (f_V - f_A\gamma_5)
e\, \bar{\nu}_L \gamma_{\mu}(1-\gamma_5)\nu_H \;,
\label{hamnd}
\end{equation}
where the form-factor $f_A$ is the subject of the present study.
The form of Eq. (\ref{hamnd}) explicates the lepton conversion 
in neutrino, rather than in the charged lepton sector.
In the next section we will present a novel contribution to the
axial form-factor $f_A$ arising from the charged current
transition at the two-loop level.
 

\section{Non-diagonal Neutrino--Electron Interaction}

The advent of the neutrino mass seems to be ``the first tangible
deviation of Nature from the original standard model'' \cite{Ram99}. 
Most of theories for neutrino
mass are accompanied by non-standard interactions of neutrinos with
matter, which may be confused with oscillations \cite{HuSV02}.
Such interactions, presented in \cite{HuSV02}, represent a further
generalization of the interaction (\ref{hamnd}).
In our present study we restrict to the modest extension of the
SM, where a mismatch in diagonalizing mass matrices of charged and
neutral leptons results in a 3$\times$3 flavor mixing matrix,
entering the charged lepton current interactions.

For massive neutrinos, the neutrino flavor eigenstates are mixtures
of the mass eigenstates, described by the leptonic 3$\times$3 unitary matrix
(sometimes dubbed the Maki-Nakagawa-Sakata $U_{\text{MNS}}$ matrix 
\cite{MaNS62}, in analogy to the
Cabibbo-Kobayashi-Maskawa matrix in the quark sector of the Standard model)
\begin{equation}
\left(\begin{array}{c}
\nu_e  \\ \nu_{\mu}  \\ \nu_{\tau}
\end{array}\right)
=
\left(\begin{array}{ccc}
U_{e1} & U_{e2}  & U_{e3} \\
U_{\mu 1} & U_{\mu 2}  & U_{\mu 3} \\
U_{\tau 1} & U_{\tau 2}  & U_{\tau 3}
\end{array}\right)
\left(\begin{array}{c}
\nu_1  \\ \nu_{2}  \\ \nu_{3}
\end{array}\right)
\equiv
U_{\text{MNS}}
\left(\begin{array}{c}
\nu_1  \\ \nu_{2}  \\ \nu_{3}
\end{array}\right) \;.
\end{equation}
Note that the flavor violation 
induced by the MNS mixing (similarly to the one by the CKM mixing) 
does not affect the renormalizability of the electroweak theory.
Thus, a safe evaluation of the quantum-loop corrections is possible.
Recently, we employed this framework in calculating lepton-flavor
violating annihilation of muonium \cite{EeKP00,EeKP01}.
Now we consider the radiative corrections which will correspond to
the two-photon exchange in diagram \ref{triangle}(a).

Since the MNS mixing affects only the charged current, our
starting point is the tree diagram displayed on Fig.~\ref{figtree}(a).
Its two-photon radiative corrections will result in Fig.~\ref{figtree}(b), as
the nondiagonal counterpart to Fig.~\ref{triangle}(a).

\begin{figure}
\includegraphics[scale=0.70]{figtree.eps}
\caption{Tree-level Feynman diagram (a), describing the nondiagonal 
neutrino--electron interaction, and
the photonic-loop diagram (b) induced by the
$\nu_{H}\nu_{L}\gamma\gamma$ vertex (the shaded circle).
\label{figtree}}
\end{figure}

\begin{figure}
\includegraphics[scale=0.68]{figloop.eps}
\caption{Samples of one-loop radiative corrections to the nondiagonal
neutrino--electron interaction.
\label{figloop}}
\end{figure}

Let us stress that for sufficiently heavy neutrino ($\nu_H$), 
such as the one allowed by the existing \emph{direct} experimental mass limit 
of $\sim$18 MeV for $\nu_{\tau}$, the diagrams in Fig.~\ref{figtree}
could give rise to the 
$\nu_{H}(P)\to\nu_{L}(p)e^{+}(k_{+})e^{-}(k_{-})$ decay.
This decay was originally considered in \cite{Sh81}, used for
constraining the $|U_{e3}|$ MNS matrix element in \cite{Ha95},
and reconsidered more recently in \cite{HoKMP99}.
However, the recent measurements squeeze neutrinos to sub-eV
mass eigenstates that exclude this decay.
Accordingly, we are left with the explicated scattering diagrams.

The referent tree-level amplitude corresponding to the diagram on
Fig.~\ref{figtree}(a) reads (after a Fierz transformation)
\begin{eqnarray}
 \mathcal{A}_{\text{tree}} &=& \frac{G_F}{\sqrt{2}} \sum_{\alpha=\mu,\tau}
  \lambda_\alpha \bar{u}(p)\gamma^{\mu}(1-\gamma_5)u(P) \nonumber \\
  &&\times\bar{u}(k_-)\gamma_{\mu}(1-\gamma_5)v(k_+) \;,
\label{tree}
\end{eqnarray}
where the summation over combinations $\lambda_\alpha \equiv 
U_{\alpha H}^{*}U_{\alpha L}$
appears on account of the unitarity of $U_{\text{MNS}}$. 
Concerning the radiative corrections,
we refer to  \cite{HoKMP99} for
a more complete exposition of the electroweak W-box diagrams 
(that are power suppressed at low energies), and the Z-triangle diagrams 
(that are suppressed by the $q^2/M_Z^2$ factor).
We explicate only the one-loop (1L) radiative contribution on 
Fig.~\ref{figloop}(a) that dominates in the set of electroweak
diagrams considered in \cite{HoKMP99}.
The pertinent amplitude
\begin{eqnarray}
 \mathcal{A}_{\text{rad}}^{\text{1L}} &=& \frac{G_F}{\sqrt{2}} 
  \frac{e^2}{24 \pi^2}
\left[ \sum_{\alpha=\mu,\tau} \lambda_\alpha 
\ln\frac{m_{\alpha}^2}{m_{e}^2}\right]
\bar{u}(p)\gamma^{\mu}(1-\gamma_5)u(P) \nonumber \\
   &&\times  \bar{u}(k_-)\gamma_{\mu}v(k_+) \;,
\label{1L}
\end{eqnarray}
contains purely vector electron current. 
This is in contrast to the contribution
displayed on Fig.~\ref{figloop}(b) that
preserves the pure V-A structure of the tree amplitude (\ref{tree}), and can
accordingly be absorbed by the Fermi coupling.  
Thus, in the sum of (\ref{tree}) and (\ref{1L})
\begin{eqnarray}
 \mathcal{A}_{\text{tree}} + \mathcal{A}_{\text{rad}}^{\text{1L}}
& = & \frac{G_F}{\sqrt{2}}
 \bar{u}(p)\gamma^{\mu}(1-\gamma_5)u(P) \nonumber \\
  & &\times  \bar{u}(k_-)\gamma_{\mu}(f_{V}- f_{A}\gamma_5)v(k_+) \;,
\label{tree1L}
\end{eqnarray}
the one-loop radiative correction on Fig. \ref{figloop}(a) modifies 
only the {\em vector} form factor
\begin{equation}
  f_V  =  \sum_{\alpha=\mu,\tau} \lambda_\alpha (1+f^{\text{1L}}_\alpha) \;,
\label{gV}
\end{equation}
leaving $f_{A}=1$ intact.
Thereby, the correction term in eq. (\ref{gV})  acquires a simple leading
logarithmic form,
\begin{equation}
  f^{\text{1L}}_\alpha  =  
\frac{\alpha}{3\pi}\ln\frac{m_{\alpha}}{m_{e}} \label{gV1L}\;.
\end{equation}

Now we focus to radiative corrections indicated on
Fig.~\ref{figtree}(b), which will give a contribution to the 
{\em axial-vector} form factor $f_A$ in (\ref{tree1L}).
This contribution corresponds to the two-loop electroweak diagrams
considered by us \cite{EeKP98} in the context of the fla\-vo\-ur-chan\-ging
$s\bar{d}\to\mu^+ \mu^-$ transitions.
In the neutrino transition at hand, we employ the
one-particle-irreducible diagrams in 't Hooft-Feynman gauge, displayed
on Fig.~\ref{fig1PI}.  These diagrams replace the shaded blob on
Fig.~\ref{figtree}(b).
\begin{figure}
\includegraphics[scale=0.65]{pilong.eps}
\caption{Electroweak one-loop Feynman diagrams contributing to
$\nu_{H}\to\nu_{L}\gamma\gamma$ in the 't Hooft-Feynman gauge.
\label{fig1PI}}
\end{figure}
Table~\ref{tblA} displays the resulting contributions
denoted by the respective insertions
enumerated on Fig.~\ref{fig1PI}. They build up the two-loop
radiative amplitude
\begin{eqnarray}
 \mathcal{A}_{\text{rad}}^{\text{2L}} &=& \frac{G_F}{4\sqrt{2}} 
  \frac{9}{4}\frac{\alpha^2}{\pi^2}
  \sum_{\alpha=\mu,\tau} \lambda_\alpha A_{(\alpha,e)}
\bar{u}(p)\gamma^{\mu}(1-\gamma_5)u(P) \nonumber \\
  && \times  \bar{u}(k_-)\gamma_{\mu}\gamma_5 v(k_+)  \;,
\end{eqnarray}
which modifies the axial-vector form factor $f_A$. The net result
\begin{equation}
  f_A  =  \sum_{\alpha=\mu,\tau} \lambda_\alpha (1 + 
\frac{9 \alpha^2}{16 \pi^2}A_{(\alpha,e)}) \;, \\
\label{urgA}
\end{equation}
is expressed in terms of the GIM-like combinations $A_{(\mu,e)}$ and 
$A_{(\tau,e)}$, displayed in Table \ref{tblA}.

\begin{table}
\caption{Contributions $A1,\ldots,A3b$ from Fig.~\ref{fig1PI} leading
to the pertinent GIM-like loop-diagram factors $A_{(\tau,e)}$ and
$A_{(\mu,e)}$.
\label{tblA}}
\begin{ruledtabular}
\begin{tabular}{cD{.}{.}{-1}D{.}{.}{-1}}
Diagram  & \multicolumn{1}{c}{$A_{(\tau,e)}$ } &
   \multicolumn{1}{c}{$A_{(\mu,e)}$}\\
\hline
  A1    &  21.4     &    14.0     \\
  A1b   &  -0.002   &    \sim 0     \\
  A2    &  0.0007   &    \sim 0     \\
  A2b    &  0.12   &     0.0007     \\
  A2c    &  0.009   &    \sim 0     \\
  A3    &  0.4   &       0.0001     \\
  A3b    &  0.03   &    \sim 0     \\
\hline 
Total   &  21.6     &   14.0 \\
\end{tabular}
\end{ruledtabular}
\end{table}

Since the dominant contribution comes from the first diagram
in Fig.~\ref{fig1PI}, one can rely on the simple leading-log
analytical form of these
functions ($A_{(\alpha,e)}\propto f_{\alpha}^{\text{2L}}$) deduced previously 
\cite{VoS76,EeKP98}. In close analogy to the one-loop radiative
correction in (\ref{gV}) and (\ref{gV1L}), our two-loop radiative
correction reads
\begin{eqnarray}
  f_A & = & \sum_{\alpha=\mu,\tau} \lambda_\alpha 
(1+f^{\text{2L}}_\alpha) \;, \label{gA}\\
  f^{\text{2L}}_\alpha & = & \frac{3}{4}\frac{\alpha^2}{\pi^2}
\ln\frac{m_{\alpha}^2}{m_{e}^2} \label{gA2L}\;.
\end{eqnarray}
The dominant $\tau$-loop ($\alpha=\tau$) corrections to the referent tree-loop
amplitude, given by expressions (\ref{gV1L}) and (\ref{gA2L}), are
numerically
\begin{equation}
 f^{\text{1L}}_\tau\simeq 6.3\cdot 10^{-3} \;, \quad
f^{\text{2L}}_\tau \simeq 6.6\cdot 10^{-5} \;.
\label{1L2L}
\end{equation}
In order to estimate $f_V$ and  $f_A$, one has to include also the
MNS-matrix prefactors, for which we now have the first strong experimental
hints.

\section{Conclusion}

  We have reached the time where the existing neutrino experiments
enable quite accurate global fits on the MNS matrix elements. Unanticipated
outcome of these experiments are large mixings. Let us, for
definiteness, display the result of an analysis by Fukugita and Tanimoto
\cite{FuT01}, based on the recent solar (SNO \cite{SNO01})
 and atmospheric (SuperKamiokande \cite{SK98}) neutrino data.

The solution corresponding to the vacuum oscillation with a 
large mixing angle (LMA)
solution to the solar neutrino problem (where only moduli of the elements can 
be shown)
\begin{equation}
\setlength{\arraycolsep}{6pt}
 U_{\text{MNS}}= \left(
\begin{array}{ccc}
0.74-0.90 & 0.45-0.65  & <0.16 \\
0.22-0.61 & 0.46-0.77  & 0.57-1/\sqrt{2} \\
0.14-0.55 & 0.36-0.68  & 1/\sqrt{2}-0.82 \\
\end{array}\right) \:,
\label{MNS}
\end{equation}
shows a ``democratic'' mixing, very different from the ``hierarchical''
one experienced in the quark sector.
A new generation of experiments is awaited in order to confirm this
scenario in which the flavor hierarchy problem faces a new subtlety.
Apart from an upgrade of the well known experiments, there
are some proposals for the ``near site'' detectors at future neutrino
factories \cite{MaY92,GGVC96,HuSV02}.


Supplied with the MNS-matrix entries, we are able to complete the
calculation of the axial form-factor (\ref{gA}),
resulting in a general structure
\begin{equation}
 U_{\mu 3}U^{*}_{\mu L}(1+ f_{\mu}^{\rm 2L})+
 U_{\tau 3}U^{*}_{\tau L}(1+f_{\tau}^{\rm 2L}) \;.
\label{UplU}
\end{equation}
Indicated two-loop contributions $f_{\alpha}^{\rm 2L}$, 
displayed in Table \ref{tblA} and comprised by the simple
relation (\ref{gA2L}), represent the novel contribution in the literature.

Some comments on the astrophysical and cosmological relevance of our
calculation are in order.  First, let us note that the processes
considered in some other accounts \cite{McFaMMR01} can also be subsumed
into the shaded blobs of Fig.~\ref{triangle}(a) or \ref{figtree}(b).
Although there are some proposals for laboratory test of radiatively
stimulated conversion of neutrinos \cite{MaY92,GGVC96}, it is
conceivable that such processes may find
their proper place in hot astrophysical
or cosmological environments.

Second, it is known that massive neutrinos may have
important implications in astrophysics and cosmology (for a review see
\cite{Ra96} and its compact update \cite{Ra99}). In particular, the
present study might be relevant for the  neutrino conversion
in astrophysical environments like supernovae, where neutrinos
play an important role.  

As noted some time ago \cite{Ra99} the
bremsstrahlung $\nu_{H}\to\nu_{L}e^{+}e^{-}\gamma$ might provide an 
important mechanism for supernovae cooling.
In view of small neutrino masses excluding this decay, the attention 
turns to related
viable mechanisms. For example, the environment might provide a
sufficient energy for the process $\nu\gamma\to\nu e^+ e^-$
which has been shown \cite{MaR00}
to become dominant at energies above $m_e$.
The Hamiltonian (\ref{hamnd})
(where the form-factors $f_V$ and $f_A$ are given by
Eqs. (\ref{gV})--(\ref{gV1L}) and (\ref{urgA})--(\ref{gA2L}), respectively)
accounts for this process, which will be considered elsewhere.

Recent consideration of the nondiagonal 
$\nu_{H}\leftrightarrow\nu_{L}$ neutrino conversion
in hot media \cite{As01} resides on the variant of our interaction 
(\ref{hamnd}) with $f_V = f_A =1$. Although we present only small
departures from this purely V-A interaction, it is conceivable
\cite{GrS98} that their effects might be considerably amplified
(the MSW resonant oscillations \cite{Wo78,MiS86} being a well known example).

Finally, a possible rendezvous of the anomaly of Fig.~\ref{triangle}(b) and its
charged current counterpart on Fig.~\ref{figtree}(b) has to be
explored yet. In the low energy world these two pieces appear as
completely decoupled. The last one is presented by the axial
form-factor in (\ref{hamnd}), having a
simple leading-log behavior and describing the neutrino 
transition in SM, as probed by photons. In this sense, our interaction 
(\ref{hamnd}) provides a starting point for some
future considerations of the neutrino
electromagnetic properties beyond the SM.
In the next step we would also like to discriminate between the
radiative correction effects and effects of non-standard interactions,
which arise in models attempting to explain neutrino masses.

\begin{acknowledgments}
I.P. thanks E. Akhmedov, J.O. Eeg and 
M. Fukugita for enlightening discussions. We acknowledge support
by the Croatian Ministry of Science (contract 119222) and by the
Research Council of Norway.
\end{acknowledgments}

\begin{thebibliography}{26} %Created by BiBTeX
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Fukuda et~al.}(1998)}]{SK98}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Fukuda}} \bibnamefont{et~al.}
  (\bibinfo{collaboration}{Super-Kamiokande}), \bibinfo{journal}{Phys. Rev.
  Lett.} \textbf{\bibinfo{volume}{81}}, \bibinfo{pages}{1562}
  (\bibinfo{year}{1998}), \eprint.

\bibitem[{\citenamefont{Ahmad et~al.}(2001)}]{SNO01}
\bibinfo{author}{\bibfnamefont{Q.~R.} \bibnamefont{Ahmad}} \bibnamefont{et~al.}
  (\bibinfo{collaboration}{SNO}), \bibinfo{journal}{Phys. Rev. Lett.}
  \textbf{\bibinfo{volume}{87}}, \bibinfo{pages}{071301}
  (\bibinfo{year}{2001}), \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Duda et~al.}(2001)\citenamefont{Duda, Gelmini, and
  Nussinov}}]{DuGN01}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Duda}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Gelmini}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Nussinov}}
  (\bibinfo{year}{2001}), \eprint.

\bibitem[{\citenamefont{Gounaris et~al.}(2001)\citenamefont{Gounaris, Paschos,
  and Porfyriadis}}]{GuPP01}
\bibinfo{author}{\bibfnamefont{G.~J.} \bibnamefont{Gounaris}},
  \bibinfo{author}{\bibfnamefont{E.~A.} \bibnamefont{Paschos}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{P.~I.}
  \bibnamefont{Porfyriadis}} (\bibinfo{year}{2001}), \eprint.

\bibitem[{\citenamefont{Rosenberg}(1963)}]{Ro63}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Rosenberg}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{129}},
  \bibinfo{pages}{2786} (\bibinfo{year}{1963}).

\bibitem[{\citenamefont{Ramond}(1999)}]{Ram99}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Ramond}},
  \emph{\bibinfo{title}{Journeys beyond the standard model}}
  (\bibinfo{publisher}{Reading, Mass., Perseus Books}, \bibinfo{year}{1999}).

\bibitem[{\citenamefont{Huber et~al.}(2002)\citenamefont{Huber, Schwetz, and
  Valle}}]{HuSV02}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Huber}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Schwetz}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~W.~F.} \bibnamefont{Valle}}
  (\bibinfo{year}{2002}), \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Maki et~al.}(1962)\citenamefont{Maki, Nakagawa, and
  Sakata}}]{MaNS62}
\bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Maki}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Nakagawa}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Sakata}},
  \bibinfo{journal}{Prog. Theor. Phys.} \textbf{\bibinfo{volume}{28}},
  \bibinfo{pages}{870} (\bibinfo{year}{1962}).

\bibitem[{\citenamefont{Eeg et~al.}(2000)\citenamefont{Eeg, Kumeri\v{c}ki, and
  Picek}}]{EeKP00}
\bibinfo{author}{\bibfnamefont{J.~O.} \bibnamefont{Eeg}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Kumeri\v{c}ki}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Picek}},
  \bibinfo{journal}{Eur. Phys. J.} \textbf{\bibinfo{volume}{C17}},
  \bibinfo{pages}{163} (\bibinfo{year}{2000}),
  \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Eeg et~al.}(2001)\citenamefont{Eeg, Kumeri\v{c}ki, and
  Picek}}]{EeKP01}
\bibinfo{author}{\bibfnamefont{J.~O.} \bibnamefont{Eeg}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Kumeri\v{c}ki}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Picek}},
  \bibinfo{journal}{Fizika} \textbf{\bibinfo{volume}{B}}
  (\bibinfo{year}{2001}), \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Shrock}(1981)}]{Sh81}
\bibinfo{author}{\bibfnamefont{R.~E.} \bibnamefont{Shrock}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{D24}},
  \bibinfo{pages}{1275} (\bibinfo{year}{1981}).

\bibitem[{\citenamefont{Hagner et~al.}(1995)\citenamefont{Hagner, Altmann, von
  Feilitzsch, Oberauer, Declais, and Kajfasz}}]{Ha95}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Hagner}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Altmann}},
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{von Feilitzsch}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Oberauer}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Declais}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Kajfasz}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{D52}},
  \bibinfo{pages}{1343} (\bibinfo{year}{1995}).

\bibitem[{\citenamefont{Ho-Kim et~al.}(2000)\citenamefont{Ho-Kim, Machet, and
  Pham}}]{HoKMP99}
\bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Ho-Kim}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Machet}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{X.~Y.} \bibnamefont{Pham}},
  \bibinfo{journal}{Eur. Phys. J.} \textbf{\bibinfo{volume}{C13}},
  \bibinfo{pages}{117} (\bibinfo{year}{2000}), \eprint.

\bibitem[{\citenamefont{Eeg et~al.}(1998)\citenamefont{Eeg, Kumeri\v{c}ki, and
  Picek}}]{EeKP98}
\bibinfo{author}{\bibfnamefont{J.~O.} \bibnamefont{Eeg}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Kumeri\v{c}ki}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Picek}},
  \bibinfo{journal}{Eur. Phys. J.} \textbf{\bibinfo{volume}{C1}},
  \bibinfo{pages}{531} (\bibinfo{year}{1998}), \eprint.

\bibitem[{\citenamefont{Voloshin and Shabalin}(1976)}]{VoS76}
\bibinfo{author}{\bibfnamefont{M.~B.} \bibnamefont{Voloshin}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{E.~P.} \bibnamefont{Shabalin}},
  \bibinfo{journal}{Pisma Zh. Eksp. Teor. Fiz.} \textbf{\bibinfo{volume}{23}},
  \bibinfo{pages}{123} (\bibinfo{year}{1976}).

\bibitem[{\citenamefont{Fukugita and Tanimoto}(2001)}]{FuT01}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Fukugita}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Tanimoto}},
  \bibinfo{journal}{Phys. Lett.} \textbf{\bibinfo{volume}{B515}},
  \bibinfo{pages}{30} (\bibinfo{year}{2001}),
  \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Matsuki and Yamamoto}(1992)}]{MaY92}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Matsuki}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Yamamoto}},
  \bibinfo{journal}{Phys. Lett.} \textbf{\bibinfo{volume}{B289}},
  \bibinfo{pages}{194} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Gonzalez-Garcia
  et~al.}(1996)\citenamefont{Gonzalez-Garcia, Vannucci, and
  Castromonte}}]{GGVC96}
\bibinfo{author}{\bibfnamefont{M.~C.} \bibnamefont{Gonzalez-Garcia}},
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Vannucci}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Castromonte}},
  \bibinfo{journal}{Phys. Lett.} \textbf{\bibinfo{volume}{B373}},
  \bibinfo{pages}{153} (\bibinfo{year}{1996}),
  \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{McFarland et~al.}(2001)\citenamefont{McFarland,
  Melissinos, Mikheev, and Repko}}]{McFaMMR01}
\bibinfo{author}{\bibfnamefont{K.~S.} \bibnamefont{McFarland}},
  \bibinfo{author}{\bibfnamefont{A.~C.} \bibnamefont{Melissinos}},
  \bibinfo{author}{\bibfnamefont{N.~V.} \bibnamefont{Mikheev}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{W.~W.} \bibnamefont{Repko}}
  (\bibinfo{year}{2001}), \eprint[http://arXiv.org/abs].

\bibitem[{\citenamefont{Raffelt}(1996)}]{Ra96}
\bibinfo{author}{\bibfnamefont{G.~G.} \bibnamefont{Raffelt}},
  \emph{\bibinfo{title}{Stars as laboratories for fundamental physics: The
  astrophysics of neutrinos, axions, and other weakly interacting particles}}
  (\bibinfo{publisher}{University of Chicago Press}, \bibinfo{year}{1996}).

\bibitem[{\citenamefont{Raffelt}(1999)}]{Ra99}
\bibinfo{author}{\bibfnamefont{G.~G.} \bibnamefont{Raffelt}},
  \bibinfo{journal}{Ann. Rev. Nucl. Part. Sci.} \textbf{\bibinfo{volume}{49}},
  \bibinfo{pages}{163} (\bibinfo{year}{1999}), \eprint.

\bibitem[{\citenamefont{Masso and Rota}(2000)}]{MaR00}
\bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Masso}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Rota}},
  \bibinfo{journal}{Phys. Lett.} \textbf{\bibinfo{volume}{B488}},
  \bibinfo{pages}{326} (\bibinfo{year}{2000}), \eprint.

\bibitem[{\citenamefont{Asida et~al.}(2001)\citenamefont{Asida, Niegawa, Ozaki,
  and Kubota}}]{As01}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Asida}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Niegawa}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Ozaki}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kubota}}
  (\bibinfo{year}{2001}), \eprint.

\bibitem[{\citenamefont{Grasso and Semikoz}(1999)}]{GrS98}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Grasso}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Semikoz}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{D60}},
  \bibinfo{pages}{053010} (\bibinfo{year}{1999}), \eprint.

\bibitem[{\citenamefont{Wolfenstein}(1978)}]{Wo78}
\bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Wolfenstein}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{D17}},
  \bibinfo{pages}{2369} (\bibinfo{year}{1978}).

\bibitem[{\citenamefont{Mikheev and Smirnov}(1986)}]{MiS86}
\bibinfo{author}{\bibfnamefont{S.~P.} \bibnamefont{Mikheev}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~Y.} \bibnamefont{Smirnov}},
  \bibinfo{journal}{Nuovo Cim.} \textbf{\bibinfo{volume}{C9}},
  \bibinfo{pages}{17} (\bibinfo{year}{1986}).

\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%% END of file nnee.tex %%%%%%%%%%%%%%

